TITLE: Prove that $\sum_{n=1}^{\infty}\ a_n^2$ is convergent if $\sum_{n=1}^{\infty}\ a_n$ is absolutely convergent
QUESTION [6 upvotes]: Suppose that $\displaystyle\sum_{n=1}^{\infty}\ a_n$ is absolutely convergent. How can we prove that $\displaystyle\sum_{n=1}^{\infty}\ a_n^2$ is convergent?

REPLY [10 votes]: $$\lim_{n \to \infty} \frac{a^2_n}{|a_n|} = \lim_{n \to \infty} |a_n| = 0,$$
since $\sum |a_n|$ converges.  By the limit comparison test, $\sum a^2_n$ converges.